If three points A, B and C with position vectors `hat(i) + x hat(j) +3 hat(k) , 3 hat(i) + 4 hat(j) + 7hat( k) ` and `y hat(i) - 2 hat(j) - 5 hat(k)` respectively are collinear, then (x,y) =

Options

A. `(-2,3)`
B. `(2,-3)`
C. `(-2,-3)`
D. `(-3,2)`

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) such that the points A, B, and C are collinear. We will use the concept of vectors and the condition for collinearity.

### Step-by-Step Solution:

1. **Identify the position vectors:**
- The position vector of point A is \( \mathbf{A} = \hat{i} + x \hat{j} + 3 \hat{k} \).
- The position vector of point B is \( \mathbf{B} = 3 \hat{i} + 4 \hat{j} + 7 \hat{k} \).
- The position vector of point C is \( \mathbf{C} = y \hat{i} - 2 \hat{j} - 5 \hat{k} \).

2. **Find the vector \( \mathbf{BA} \):**
\[ \mathbf{BA} = \mathbf{A} - \mathbf{B} = \left( \hat{i} + x \hat{j} + 3 \hat{k} \right) - \left( 3 \hat{i} + 4 \hat{j} + 7 \hat{k} \right) \]
\[ = (1 - 3) \hat{i} + (x - 4) \hat{j} + (3 - 7) \hat{k} = -2 \hat{i} + (x - 4) \hat{j} - 4 \hat{k} \]

3. **Find the vector \( \mathbf{BC} \):**
\[ \mathbf{BC} = \mathbf{C} - \mathbf{B} = \left( y \hat{i} - 2 \hat{j} - 5 \hat{k} \right) - \left( 3 \hat{i} + 4 \hat{j} + 7 \hat{k} \right) \]
\[ = (y - 3) \hat{i} + (-2 - 4) \hat{j} + (-5 - 7) \hat{k} = (y - 3) \hat{i} - 6 \hat{j} - 12 \hat{k} \]

4. **Set the vectors \( \mathbf{BA} \) and \( \mathbf{BC} \) to be parallel:**
Since A, B, and C are collinear, \( \mathbf{BA} \) and \( \mathbf{BC} \) must be parallel. Therefore, we can write:
\[ \mathbf{BC} = \lambda \mathbf{BA} \]
for some scalar \( \lambda \).

5. **Equate the components:**
From the equality of the vectors, we have:
\[ (y - 3) \hat{i} - 6 \hat{j} - 12 \hat{k} = \lambda (-2 \hat{i} + (x - 4) \hat{j} - 4 \hat{k}) \]
This gives us the following equations:
- For the \( \hat{i} \) components: \( y - 3 = -2\lambda \)
- For the \( \hat{j} \) components: \( -6 = \lambda (x - 4) \)
- For the \( \hat{k} \) components: \( -12 = -4\lambda \)

6. **Solve for \( \lambda \):**
From the third equation:
\[ -12 = -4\lambda \implies \lambda = 3 \]

7. **Substitute \( \lambda \) back to find \( x \):**
Substitute \( \lambda = 3 \) into the second equation:
\[ -6 = 3(x - 4) \implies -6 = 3x - 12 \implies 3x = 6 \implies x = 2 \]

8. **Substitute \( \lambda \) back to find \( y \):**
Substitute \( \lambda = 3 \) into the first equation:
\[ y - 3 = -2(3) \implies y - 3 = -6 \implies y = -3 \]

9. **Final Answer:**
Thus, the values of \( (x, y) \) are \( (2, -3) \).

### Summary:
The final answer is \( (x, y) = (2, -3) \).